Question

Please help me as soon as posable!! In hurry!!(24 points)

In a geometric sequence, a_ 2 = 2, a_ 3 = 20, and a_4 = 200.
Which equation can be used to find the nth term of the sequence, a_n?

A) a_n =2^n-1
B) a_n =2 · 18^n-1
C) a_n =10 · 2^n-1
D) a_n =1/5 · 10^n-1

Answer 1

The sequence is geometric, so

`a_n = r a_(n-1)`

for some constant r. From this rule, it follows that

`a_3 = r a_2 \implies 20 = 2r \implies r = 10`

and we can determine the first term to be

`a_2 = r a_1 \implies 2 = 10 a_1 \implies a_1 = \frac15`

Now, by substitution we have

`a_n = r a_(n-1) = r^2 a_(n-2) = r^3 a_(n-3) = \cdots`

and so on down to (D)

`a_n = r^(n-1) a_1 = 10^(n-1) \cdot \frac15`

(notice how the exponent on r and the subscript on a add up to n)